I recently graduated from a Masters in mathematical physics. During my time I especially enjoyed pure maths courses and picked as many as I could. Since graduating I have been exploring my options and am intrigued by the subject of elliptic curves. I am trying to build a background in my spare time, with the goal of writing a short presentation of the topic, to learn by teaching and to potentially use in future applications.

If I continue enjoying the subject, I would like to apply for a PhD. I feel I should get some point of contact with potential supervisors as soon as possible, however it feels rude to do so with only a very vague understanding of their work.

My question

How do I initiate contact with academics working on elliptic curves despite being very new to the subject. Could I ask them to explain something that would be found in a textbook ? For example, the Tate-Shafarevich group seems to be quite actively researched. Could I ask them why they're interested in this group ?

  • 3
    I recently graduated from a Masters in mathematical physics please note that a degree shows (simply) that you can understand answer to already existing problems. Final goal of a PhD is to present answer to existing, unsolved, problems. If you do not show independence in understanding already existing solutions/approach (i.e. the example group you bring) no one can productively support you in your quest towards searching answers to unsolved problems. In research, your Master degree is just a starting point that you can do the (necessary but not sufficient) bare minimum ...do that minimum :)
    – EarlGrey
    Oct 5, 2022 at 23:06
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    I met a mathematics post-doc once and asked her what she was working on. She squinted, thought carefully for a bit, and came up with... "Imagine you have a universe." She stopped there but I nodded attentively, so she thought about it a bit more. "Okay so, now imagine it's expanding." Me: "Uh huh?" Her: "It's kind of about that." I let her off the hook by not asking any more about it.
    – Beanluc
    Oct 6, 2022 at 18:02
  • How could asking an expert to explain anything which might be found in text-books not be seen as insulting? If you have only a vague understanding of the professor's subject why would you dream of contacting, let alone challenging the professor, rather than simply reading three or four more text-books? Oct 7, 2022 at 21:49
  • 1
    @RobbieGoodwin Asking experts which books to read is usually a good idea.
    – user153715
    Oct 8, 2022 at 0:01
  • @Beanluc: So, what was the subject she was working on?
    – Ink blot
    Oct 8, 2022 at 10:10

4 Answers 4


Here's what I would recommend.

  1. First, I would learn the basics of elliptic curves by consulting a standard book on elliptic curves. A good first option is the book by Silverman and Tate. You could also consult the book "The Arithmetic of Elliptic Curves" by Silverman. These books focus on the relationship to number theory. "Lectures on Elliptic Curves" by Cassels is also really good.

  2. After spending some time on it, if you are interested, you could look for people who work on elliptic curves. Now you have found people who think about elliptic curves. Then what?

  3. Go to their list of publications and look at some of them. You don't have to try to understand the papers completely. Just read the introductions to these papers. Try to answer for yourself: What do they do in the paper and why?

  4. Then you could send them a short e-mail (5 sentences or so). You could say the following:

a. You've been reading about elliptic curves on your own, using the references [insert the books/articles you've been reading] and have been enjoying learning about them.

b. You're interested in contemporary research in this area, and you found out that they work on this topic.

c. You've looked at some of their publications and have read the introductions to their publications to get an idea of what they work on.

d. You would like to learn more. Then ask if they could recommend something to read to help them learn about current research.

This shows that you're interested in their work, and that you've done some independent reading.

(Also, roughly speaking, the reason some people are interested in the Tate-Shafarevich group is that a lot of interesting results about arithmetic aspects of elliptic curves, like their L-functions, are only true under the assumption that the Tate-Shafarevich group is finite.)

  • Thank you for your complete answer. This answers the part I was most worried about: transitioning from learning background to finding out more about their research.
    – Mr Lolo
    Oct 5, 2022 at 11:14
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    Unless you do a masters in math, or some sort of reading course with an undergraduate professor, it's hard to know what people work on. At least in the US, I don't think you have to be familiar with a professor's research before applying. For me, I had an idea of what my interests were, but didn't pick an advisor until a year or so after I started my PhD program.
    – cgb5436
    Oct 5, 2022 at 11:50
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    @cgb5436 In most European countries, it's the exact opposite. When you start a PhD program, you start it with a specific supervisor on a specific project. Oct 6, 2022 at 8:46
  • @Carl-FredrikNybergBrodda: Though in Germany you do a Master's before the PhD, the PhD is just three years. Oct 7, 2022 at 6:42
  • @MartinUeding Not all PhD programs in Europe require a Master's degree before doing a PhD. Oct 7, 2022 at 7:41

it feels rude to do so with only a very vague understanding of their work

Actually, it is rude, if you are not enrolled in their course , however luckily we live in the year 2022 and you can access an online course to build at least a minimum understanding of the topic, since I assume you have the time and the (intellectual) resources to attend such a course, or at least to work out the syllabus of such a course.

Additionally, you may join a workshop or a conference on the topic. If you write nicely your situation (not a student, unemployed, interested in the topic, looking for opportunities for a PhD on the topic of) to the organizers, they are likely to waive the fees to access the conference (if any).

  • 9
    Have you been offended by having someone interested in your work? It might be rude to make requests, but I don't think that's what OP was referring to. Can you say more? Oct 5, 2022 at 21:21
  • @JustinMeiners if my working life is on the small details of subsubtopic Z of subtopic Y of topic X, and random [email protected] writes me asking about "may you enlighten me and provide me a learning path to subtopic Y of topic X" I am annoyed, I would either refer to public available material on X.Y, or to my personal publication on topic X.Y saying "please read carefully the reference". I would not carry a personal grudge to mr/ms [email protected], but I would spend the least time possible, and I would surely not be interested in a "future cooperation".
    – EarlGrey
    Oct 5, 2022 at 22:10
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    @JustinMeiners if, on the other hand, mr/ms [email protected] would ask me a question on sub-sub-topic Z, showing at least marginal knowledge on my publication on the topic and showing interest in 1) extending my work, or 2) applying my work, or even 3) confuting my work, then I would be very interested in having an half an hour phone call with mr/ms [email protected]
    – EarlGrey
    Oct 5, 2022 at 22:13
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    My impression is that OP wants to contact people because they're interested in working with them for a PhD degree and they're not sure how to express interest when they themselves currently lack background knowledge. They're not e-mailing just because they want to know more about some widely known fact in the literature.
    – cgb5436
    Oct 5, 2022 at 22:46
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    @cgb5436 "Could I ask them to explain something that would be found in a textbook ? [...] Could I ask them why they're interested in this [...] ?" It would be rude to ask the first question, I would be annoyed from the second question, but having enough time I would consider answering it unless this question is posed in the terms "why is it interesting and why should I (OP) be interested in this to pursue a PhD": in that case I would trash it immediately. I am not you, I have already to figure out what is interesting to me :)
    – EarlGrey
    Oct 5, 2022 at 23:01

How do I initiate contact with academics working on elliptic curves despite being very new to the subject. Could I ask them to explain something that would be found in a textbook?

Unlike the other individual points of view expressed here, I think this can certainly be done in a way that neither rude, spam, nor unwise.

While there are certainly "I can't be bothered" types in any academic field, there are also plenty of folks who understand that the finding various ways to explain their research, especially to people with different levels of familiarity is a valuable exercise for scholars and some would even see it as a sacred duty1.

Enlightened scholars will also know that sometimes unexpected benefits arise from unexpected or nonstandard discussions.

I would not say that if they refuses or ignored a well-written email of this type that they were "bad" or unenlightened, but I would see their willingness to discuss with you or not as a sort of filter for the kind of person you'd like to get to know in the first place!

Okay, but what would a "well-written email of this type" be like?

It should be honest and upfront but not off-putting. Explain that you don't want a full lecture on the topic, but before you take the dive you want a little bit of guidance and/or clarification to make sure you get started on the right foot and path. Include a few specifics that make it clear you have at least some understanding of the topic and that they would not be wasting their time.

Mention a time limit, like 15 or 20 minutes. Of course if they feel it's productive they may extend, but make it clear you have every intention of keeping it short and promptly leaving if they don't offer to extend. You want to give them an opportunity to see if they can help, without fear that they have a lot to loose.

  • If you get a positive response2, then despite what you said, study like crazy and make at least some progress so that they might be pleasantly surprised that you were a bit modest describing your level of understanding, and so that they are further reassured that spending a little time with you has a chance of making a real difference. To that end cgb5436's answer suggests some resources to start with.
  • If you get a negative response send a short, concise thank you reply and follow up on your own on any advice they might have included in their response (unless it's bad advice!)
  • If you get no response then you should not pursue it further with that person at this time. While occasionally mail messages get missed or move to the spam folder, most likely they don't want to deal with the situation, and you don't want to do anything that looks troublesome or harassing or else word of you may spread in a negative way.

  • 1in moderation.
  • 2and even if you don't!

Unless you are enrolled in their classes or you find some mistakes in their books that you would like to report, I don't think it is a good idea to contact them (that way). In the worst case, they may mark your email as spam.

If you only want to initiate the conversation with them, please don't do it this way. It is not wise to impress them like this.

If have some doubts while reading the books. Please consider asking professors whom you know. They may provide some help.

Furthermore, do your "homework". Keep reading and studying the topic yourself. At least, you need to have a decent understanding of the topic to know if you really like it. Some topics may sound super interesting at first glance, but turn out to be terribly boring later. When you are ready, prepare the necessary documents for the application. Do some research on potential supervisors and contact them (if it is appropriate)


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